The obvious problem is that a law degree in most countries outside the US is an undergraduate degree, but in the US you need to have completed an undergraduate degree to start law school.

Assuming you have an undergraduate degree already, U.S. law schools seem fairly flexible about letting people study for another degree at the same time.

I'm not sure why you would want to do this, though. Problems:

(1) it would be more efficient to complete one degree first and then start the other.

(2) having a foreign law degree would give you very little advantage if you want to practice law in the U.S.

(3) if you just want to practice law in your own country, why not just get an LLM? That's both cheaper and easier.

(4) U.S. law degrees are extremely expensive, and people educated outside the U.S. seldom get more than 1/2 tuition scholarships. Also, you wouldn't be eligible for federal loans, which is how U.S. students finance their legal education.

I have several friends who've completed the Columbia Law School/ University College London LLB/JD program. Cornell Law also has dual degree programs in France and in Germany. I don't have any experience with these programs other than getting accepted to Cornell's programs, but I think they're all worthwhile. I hope these links help.